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Cooperative Games an Introduction to Game Theory – Part Ii LARS-ÅKE LINDAHL COOPERATIVE GAMES AN INTRODUCTION TO GAME THEORY – PART II i Cooperative Games: An Introduction to Game Theory – Part II 1st edition © 2017 Lars-Åke Lindahl & bookboon.com ISBN 978-87-403-2136-4 Peer review by Prof. Erik Ekström, Professor in mathematics, Uppsala University ii COOPERATIVE GAMES: AN INTRODUCTION TO GAME THEORY – PART II CONTENTS CONTENTS Non-Cooperative Games: An Introduction to Game Theory – Part I 1 Utility Theory Part I 1.1 Preference relations and utility functions Part I 1.2 Continuous preference relations Part I 1.3 Lotteries Part I 1.4 Expected utility Part I 1.5 von Neumann-Morgenstern preferences Part I 2 Strategic Games Part I 2.1 Definition and examples Part I 2.2 Nash equilibrium Part I 2.3 Existence of Nash equilibria Part I 2.4 Maxminimization Part I 2.5 Strictly competitive games Part I 3 Two Models of Oligopoly Part I 3.1 Cournot’s model of oligopoly Part I 3.2 Bertrand’s model of oligopoly Part I 4 Congestion Games and Potential Games Part I 4.1 Congestion games Part I 4.2 Potential games Part I 5 Mixed Strategies Part I 5.1 Mixed strategies Part I 5.2 The mixed extension of a game Part I 5.3 The indifference principle Part I 5.4 Dominance Part I 5.5 Maxminimizing strategies Part I 6 Two-person Zero-sum Games Part I 6.1 Optimal strategies and the value Part I 6.2 Two-person zero-sum games and linear programming Part I 7 Rationalizability Part I 7.1 Beliefs Part I 7.2 Rationalizability Part I iii COOPERATIVE GAMES: AN INTRODUCTION TO GAME THEORY – PART II CONTENTS 8 Extensive Games with Perfect Information Part I 8.1 Game trees Part I 8.2 Extensive form games Part I 8.3 Subgame perfect equilibria Part I 8.4 Stackelberg duopoly Part I 8.5 Chance moves Part I 9 Extensive Games with Imperfect Information Part I 9.1 Basic Endgame Part I 9.2 Extensive games with incomplete information Part I 9.3 Mixed strategies and behavior strategies Part I Answers and hints for the exercises Part I Index Part I Cooperative Games: An Introduction to Game Theory – Part II Preface v 10 Coalitional Games 1 10.1 Definition 1 10.2 Imputations 4 10.3 Examples 7 10.4 The core 13 10.5 Games with nonempty core 19 10.6 The nucleolus 28 11 The Shapley Value 43 11.1 The Shapley solution 43 11.2 Alternative characterization of the Shapley value 51 11.3 The Shapley-Shubik power index 58 12 Coalitional Games without Transferable Utility 62 12.1 Coalitional games without transferable utility 62 12.2 Exchange economies 65 12.3 The Nash bargaining solution 72 Appendix 1: Convexity 80 Appendix 2: Kakutani’s fixed point theorem 83 Brief historical notes 87 Answers and hints for the exercises 91 Index 93 iv COOPERATIVE GAMES: AN INTRODUCTION TO GAME THEORY – PART II PREFACE Preface Non-cooperative game theory focuses on the individual players’ strategies and their influence on payoffs and tries to predict what strategies the players will choose. It asks how people should act. Cooperative game theory, on the other hand, abstracts from the individual players’ strategies and instead focuses on the coalitions players may form. Cooperative games can be seen as a competition between coalitions of players, rather than between individual players. The big advantage of the cooperative theory is that it does not need a precisely defined structure for the actual game. It is enough to say what each coalition can achieve; you need not say how. The basic assumption is that the grand coalition, that is the group consisting of all players, will form, and the main question is how to allocate in some fair way the payoff of the grand coalition among the players. The answer to this question is a solution concept which, roughly speaking, is a vector that represents the allocation to each player. Different solution concepts based on different notions of fairness have been proposed, and we will study three of them in this volume, namely the core, the nucleolus and the Shapley solution. This Part II of An Introduction to Game Theory is essentially indepen- dent of Part I on Non-Cooperative Games and references to results in Part I only appear in a few places. Part II can therefore very well be read and studied before Part I. Lars-Ake˚ Lindahl 1 v COOPERATIVE GAMES: AN INTRODUCTION TO GAME THEORY – PART II COALITIONAL GAMES Chapter 10 Coalitional Games In many situations with several people involved, the total result of their efforts is best when everyone cooperates, and conflicts of interest only arise when the collective profit is to be distributed among the individuals. In this chapter we will model such situations as games. We assume that all players use the same unit when measuring their utilities, and that the total payoff that a group can achieve through cooperation can be distributed freely among the members of the group. The main problem is to determine how the payoff should be distributed, and we will study some different solutions. 10.1 Definitions Definition 10.1.1 A coalitional game N,v (with transferable utility) con- sists of a finite set N of players and a real-valued function v, defined on the set of all nonempty subsets of N. C The sets in , i.e. the nonempty subsets of N, are called coalitions, and the function valueC v(S) of a coalition S is called the coalition’s value. The function value v(N) is the game’s total value. The number of members of a coalition S will be denoted by S . | | In order for some definitions and induction proofs to work, we sometimes need to extend the value function v so that it is also defined for the empty set , which we always do by defining v( ) = 0. ∅ ∅ The entire set N is a coalition, the so called grand coalition. The players in N will in general be numbered 1, 2, . , n, where n = N . The number | | of coalitions, i.e. nonempty substs of N,is2n 1. − The intuitive interpretation of the value v(S) is that it is the total utility, wealth or power, that players in the S-coalition can achieve by cooperating, 1 1 COOPERATIVE GAMES: AN INTRODUCTION TO GAME THEORY – PART II COALITIONAL GAMES 2 10 Coalitional Games regardless of what players outside the coalition do. In particular, v(N) is the total utility obtained when all players cooperate. In order to derive interesting results, we need to constrain the class of games through appropriate restrictions on the value function. In the following definitions, two important subclasses are defined. Definition 10.1.2 A coalitional game N,v is called cohesive if v(S )+v(S )+ + v(S ) v(N) 1 2 ··· k ≤ for every partition S ,S ,...,S of N. { 1 2 k} Recall, that a partition of a set M is a family of pairwise disjoint subsets whose union is equal to M. The players of a cohesive game profit from keeping together, which ex- plains the term ”cohesive”. It is impossible to create greater total utility by splitting the grand coalition into a number of subgroups. In particular, v(S)+v(N S) v(N) \ ≤ for all coalitions S, and v( i ) v(N). { } ≤ i N ∈ Cohesiveness will be a natural prerequisite for many results in this chapter. Superadditive games form an important subclass of cohesive games and are defined as follows. Definition 10.1.3 A coalitional game is called superadditive if v(S)+v(T ) v(S T ) ≤ ∪ for all pairwise disjoint coalitions S and T . Superadditivity implies cohesiveness, but the converse is not true. Example 10.1.1 The coalitional game N,v with N = 1, 2, 3 and value function { } v( 1 )=0,v( 2 )=v( 3 )=2,v( 1, 2 )=v( 1, 3 )=v( 2, 3 )=3, { } { } { } { } { } { } v( 1, 2, 3 )=5 { } is cohesive, because there are four partitions of N, namely 1 , 2 , 3 , 1 , 2, 3 , 2 , 1, 3 and 3 , 1, 2 , and the left side{{ of} the{ inequal-} { }} {{ity in} { Definition}} {{ 10.1.2} { is}} in turn{{ equal} { to}} 0 + 2 + 2, 0 + 3, 2 + 3 and 2 + 3, which in all cases is 5=v(N). However, the game≤ is not superadditive, because v( 2 )+v( 3 )=4> { } { } v( 2, 3 ). { } 2 COOPERATIVE GAMES: AN INTRODUCTION TO GAME THEORY – PART II COALITIONAL GAMES 10.1 Definitions 3 A coalitional game’s value function v is a primitive concept. How and in what way the value v(S) depends on the efforts of the individual players in S is irrelevant and unimportant in the context, but of course, this does not preclude us from defining a player’s marginal contribution to a coalition as follows. Definition 10.1.4 Let i be a player in the game N,v . The player’s marginal contribution to the coalition S is the quantity ∆ (S)=v(S) v(S i ). i − \{} A player’s marginal contribution to a coalition he does not belong to is of course always zero (since S = S i if i/S). A player’s marginal contribution is thus only interesting for\{ coalitions} ∈ to which he belongs. In superadditive games, ∆i(S) v( i ) for all i S. A player’s marginal contribution to a coalition to which≥ he{ belongs} is thus∈ greater than or equal to the value that the player can achieve on his own.
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